How can I distribute a 3.6 kg load over multiple nails using a string?

Question

I'm trying to solve a practical problem with what feels like should be basic physics. I've guessed that this problem might be about vectors, pulleys and trigonometry, but it's knowledge which I lack, and I'd like to learn. Asking this on the Physics forum redirected me to Engineering.

I have a 3.9 kg painting, and I'm resticted to hanging it on a wall using adhesive nails. The best adhesive nails I have access to are rated for a 2 kg load (its contact surface with the wall is 4 x 6 cm, in case distances between nails becomes relevant). I also have a string rated for a 6 kg load, and the painting's hanging mechanism works with a string.

I want to figure out a hanging "system" that distributes the painting’s weight over more than 2 nails (which would be cutting it too close for their 2 kg measurement margin of error), using a string for distribution, so that no single nail is overloaded. I have 3 questions:

Question 1: Could I calculate a theoretical optimal height for points A and D below to evenly distribute forces?

• I received a recommendation to create a cantenary arc out of nails, with a string loop around them, and the painting hanging at the bottom. Used this and after 5 days an edge nail snapped off the wall (edge D), implying either an adhesive bonding flaw or unequal force distribution. The adhesive had marks of torque on it. I may have underestimated how low A and D should be, and I'd try reattaching them at a lower height.

    B-----C
   /       \
  A---------D
       ↓
     3.6 kg
  

Question 2: Is a square configuration a sound option?

• I asked ChatGPT (o3-mini-high) just to see what answer it produced, and it came up with a square configuration. In my scientific ignorance I'd assume the top nails have greater forces applied on them than the bottom nails, but it produced an explanation I could not verify: "Even if a nail is lower, the rope still 'wraps' around it for a certain angle. That contact, along with friction between the rope and the nail, forces the rope to push against the nail. This means that each nail experiences a reaction force that helps support the weight, even if the nail is not at the very top."

  B-------C
  |       |
  |       |
  A-------D
      ↓
    3.6 kg
  

Question 3: Is there a theoretical ideal solution to this problem?

• For the purposes of this question we can assume that other variables regarding adhesives and the wall's construction are negligeable, I propose ignoring the complexity added by the rope's sagging under tension, and the practical limitations I'm working with mean there are no better nails that could be used

Answer 1

You can hang unlimited weights if you use a tree pattern.

Use as many nails as needed and connect them by string as shown on the diagram.

Edit

My apologies for not clarifying; only the top Os are nails; the others are nodes.

n   n   n   n
 \ / \ / \ /
  o   o   o
   \ / \ /
    o   o
     \ /
      o
n = nail / o = knot

Answer 2

Question 1: Could I calculate a theoretical optimal height for points A and D below to evenly distribute forces?

Yes. Or, someone could. If you know the tension on the string and the angle it turns as it goes over a nail, then you can calculate the magnitude and direction of the force on the nail.

Then you can play with the nail configurations manually to find the best fit, or if you want a deep dive into numerical analysis, you could throw the whole thing at an optimizer and let it find the best fit.

Question 2: Is a square configuration a sound option?

No. See below. The force on the bottom two points would always be inward and upward; you could relieve more stress on the top two nails by spacing the bottom out at least a bit.

Question 3: Is there a theoretical ideal solution to this problem?

I don't know. But I'm pretty sure there's a theoretically ideal condition you can place on your grouping.

Assuming a weightless, frictionless, perfectly flexible string, then you'll even out the forces on all of the adhesive patches if the string goes through the same angle as is passes over each nail. This I'm sure of. It comes from the fact that for this ideal string, the force on the nail will be equal to the tension on the string times twice the cosine of half of the angle between the string segments. So, equalize the angles, and you equalize the forces.

I'm pretty sure (nearly certain, in fact) that if you even out the forces on each nail and minimize the angle off of vertical of each end of the string where it hangs over the nails, then you'll minimize the maximum force on any one nail.

For a given width of your constellation of nails and a given drop to a single point of support on the picture, then for any given number of nails there should be just one arrangement that'll satisfy the conditions.

For that matter, if you have two points of support on the picture a known distance apart, there should still be a single constellation that'll satisfy that criterion.